The present invention relates to the art of medical diagnostic imaging. It finds particular application in conjunction with magnetic resonance imaging and will be described with particular reference thereto. It is to be appreciated, however, that the invention will also find application in other imaging applications in which data is transformed between frequency and time or spatial domains.
Because common discrete Fourier transforms are extremely slow, they are not generally used in computer data processing of spectrographic or analogous data. A data line with N data elements requires N.sup.2 mathematical operations. In the mid 1960's, the Cooley-Tukey algorithm was developed which performed a Fourier transform operation with only Nlog.sub.2 N mathematical operations commonly called a "fast Fourier transform". The fast Fourier transform algorithms were limited in that N was required to be an integer power of an integer known as the Radix value, most commonly 2. The dramatic increase in speed was considered more than worth the limitation of the length of the data lines. Note that for a data line with 512 samples (N=512), the discrete Fourier transform required over 260,000 mathematical operations whereas the fast Fourier transform only requires about 4,600. Because computing time is roughly proportional to the number of mathematical operations, the discrete Fourier transform required about 56 times as long as a fast Fourier transform to process a 512 sample line. Due to the exponent in this relationship, larger data lines achieved an even more dramatic time savings. The fast Fourier transform reduced the computing time sufficiently that fast Fourier transforms became a standard computer subroutine.
In magnetic resonance imaging in which two dimensional inverse Fourier transforms are utilized, the time savings is even more dramatic. Although magnetic resonance literature often refers to a "Fourier transform", those skilled in the art understand that a fast inverse Fourier transform computer subroutine is utilized in practice. The universal use of the fast Fourier transform is also evidenced by image sizes that are integer powers of two, such as 512.times.512, 256.times.256, etc.
The use of square image matrices is dictated by the integer power of 2 requirements of the fast Fourier transform algorithm. In medical diagnostic imaging, the regions of a human body to be imaged are rarely square. Rather, the region of interest through the torso tends to be relatively short and wide; whereas, a region of interest through the skull tends to be relatively high and narrow. To accommodate the demands of the fast Fourier transform algorithm, the field of view has commonly been selected to match the major dimension of the imaged area. The minor dimension normally being smaller than the field of view, was oversampled.
One technique for fitting a rectangular or oval patient section into a square matrix is illustrated in U.S. Pat. No. 4,748,411. In an example in which the patient fills only two thirds of the field of view, only two thirds the normal number of data lines would be collected. To accommodate the fast Fourier transform algorithm demands for a data set in which the number of samples is equal to an integer-power-of-two, additional zeros were added to the data set so that it was again a 256.times.256 or other integer-power-of-two square.
In the fast Fourier transform, the added zeros are a form of interpolation. In the resultant 256.times.256 image generated by the zero filled fast Fourier transform, the subject is stretched to fill the entire 256.times.256 square of the image. That is, along the phase encode direction, the data is stretched to 3/2 its prior size such that the full square is filled with only 2/3 the data. To remove the distortion, the '411 patent reduces the image by 2/3 to return it to its undistorted size. The resolution of the resultant contracted image is controlled by the increment in the phase encoding between adjacent phase encoded views.
The present invention provides a new and improved apparatus and method for generating improved quality images from data sets or matrices of arbitrary size.